where \(\operatorname{ind}(u, \, \partial B_\delta (a_j))\)  is a local topological invariant
— the index — computed on the boundary of a small geodesic ball \(B_\delta (a_j)\) ,
with radius \(\delta \)  and center \(a_j\) . The radius \(\delta \)  must be taken small enough,
so that the ball \(B_\delta (a_j)\)  does not contain any zero of \(u\)  other than \(a_j\) .
(For the definition of the index, see Section  below.)
An immediate consequence is that a non-vanishing continuous vector field
exists if and only if the Euler characteristic of \(M\)  is equal to zero, namely only if \(M\)  is a torus (up to homeomorphism).
In other words, if \(\chi (M)\ne 0\)  then \(M\)  does not support a globally defined, unit-norm, continuous vector field.
Interestingly, the very same topological obstruction holds for Sobolev vector fields.
More precisely (see [1]} and [2]}), indicating with \(D\)  the Levi-Civita connection, we have that the Sobolev class
\(\left\lbrace v:M\rightarrow TM: v(p)\in T_p M,\,\,\left| v \right| =1\,\,\text{a.e. in }M,\,\,\left| D v \right|_g\in L^2(M)\right\rbrace \ne \emptyset \Longleftrightarrow \chi (M)=0.\) 
